Beats
When two sounds with nearly the same frequency play together, they drift in and out of step with each other. Sometimes their vibrations add up to make a louder sound, and sometimes they cancel out to make it quieter, so the loudness pulses up and down. This pulsing rate, called the beat frequency, is simply the difference between the two frequencies.
The formula
f_beat = |f₁ − f₂|
- f_beat — beat frequency (Hz): how many times per second the loudness pulses
- f₁ — frequency one (Hz): the frequency of the first sound
- f₂ — frequency two (Hz): the frequency of the second sound
Worked example
A tuning fork rings at 442 Hz, and a guitar string rings at 438 Hz. How many beats per second will you hear?
- f₁ = 442 Hz
- f₂ = 438 Hz
- f_beat = |f₁ − f₂|
- f_beat = |442 Hz − 438 Hz|
f_beat = 4 Hz
Test yourself
A guitar string rings at 440 Hz, and a tuning fork rings at 445 Hz. How many times per second will the sound pulse louder and quieter?
- Correct answer: 5 times per second
- 885 times per second
- 440 times per second
Right! f_beat = |445 Hz − 440 Hz| = 5 Hz, so five pulses per second.
You're tuning a guitar string to match a 440 Hz reference. As you tighten the string closer to the correct pitch, what happens to the beats you hear?
- They speed up
- Correct answer: They slow down and eventually disappear
- They stay the same speed
Yes! As the frequencies get closer, the difference |f₁ − f₂| shrinks, so the beats slow down until they vanish at perfect tune.
Where you see this
Piano tuners and guitarists work by ear with this effect every day: sound the string against a reference fork, and if the two pitches are close but not equal, the volume wobbles — a wobble of a few pulses per second means almost in tune, and a wobble that slows and finally vanishes means dead on.
Common mistakes
The classic slip is computing the beat rate from the wrong combination — adding the frequencies or just quoting one of them, when the pulse rate is the difference: f_beat = |f₁ − f₂|, so 445 Hz against 440 Hz gives 5 pulses per second. The second misconception is that the sources themselves are getting louder and quieter — both ring perfectly steadily, and the pulsing is interference in time, adding and cancelling at your ear.
How it connects
Beats are superposition — two waves at slightly different frequencies sliding in and out of step, the same adding-and-cancelling that interference and diffraction perform across space in optics. Within the waves unit they show why 'almost the same frequency' is an audible condition, not an inaudible one; sound intensity, next, follows what happens to a single wave's energy as it spreads.